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The usual transformations on homogeneous coordinates for 2 $\mathrm{D}$ computer graphics involve $3 \times 3$ matrices of the form $\left[\begin{array}{cc}{A} & {\mathbf{p}} \\ {\mathbf{0}^{T}} & {1}\end{array}\right]$ where $A$ is a $2 \times 2$ matrix and $\mathbf{p}$ is in $\mathbb{R}^{2} .$ Show that such a transformation amounts to a linear transformation on $\mathbb{R}^{2}$ followed by a translation. [Hint: Find an appropriate matrix factorization involving partitioned matrices.]

$\left[\begin{array}{cc}{I} & {p} \\ {0^{T}} & {1}\end{array}\right] \cdot\left[\begin{array}{cc}{A} & {0} \\ {0^{T}} & {1}\end{array}\right]=\left[\begin{array}{cc}{A} & {p} \\ {0^{T}} & {1}\end{array}\right]$

Algebra

Chapter 2

Matrix Algebra

Section 7

Applications to Computer Graphics

Introduction to Matrices

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

02:10

An affine transformation $…

04:46

Let $T_{1}: \mathbb{R}^{2}…

02:49

Let $T : \mathbb{R}^{2} \r…

01:19

In Exercises $1-10$ , assu…

02:41

06:15

02:16

The linear transformation …

01:38

07:46

A rotation in $\mathbb{R}^…

00:53

Determine the linear trans…

so we're given that on upon transformation is one of the form t of X is equal to mu tricks, eh? Turns the vector X plus a vector b ah, where a is on by n matrix and B is on and by one vector s So we want to show that if bees non zero the tea's not a linear transformation So we need to check the condition for TV linear. So FT was Lanier ah t of X plus y would have to equal t of x plus t of Why so let's check both sides. So the left hand side t of explosive by it's just equal thio a times that's close. Why plus B and T X plus you, Why iss equal to you? A x It must be waas a by plus B. And since these things are not equal to each other, he's not letting her transformation. And also another way, Um, it's a check. If t was linear than t of zero would have to equal zero. But t of the zero factor is just equal to a times Does your elector plus B which is just the vector B. So if these non zero then t is not

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02:09

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08:45

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